Problem: The velocity of a particle moving along the $x$ -axis is $v(t)=\cos(t)$. At $t=\dfrac{\pi}{2}$, its position is $2$. What is the position of the particle, $s(t)$, at any time $t$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $s(t)=\sin(t)$ (Choice B) B $s(t)=1-\sin(t)$ (Choice C) C $s(t)=\sin(t)+1$ (Choice D) D $s(t)=\sin(t)+3$
We know that $s(t)= \int v(t) \,dt$. In this case, $s(t)= \int \cos(t) \,dt$ Let's find the indefinite integral: $\begin{aligned} \int \cos(t) \,dt&=\sin(t)+C\\ \end{aligned}$ We know that $s\left(\dfrac{\pi}{2}\right)=2$. Let's use this information to solve for $C$. $\begin{aligned}s(t)&=\sin(t)+C\\ \\ s\left(\dfrac{\pi}{2}\right)&=\sin\left(\dfrac{\pi}{2}\right)+C\\ \\\\ 2&=1+C\\ \\ 1&=C \end{aligned}$ The position of the particle $s(t)$ at any time $t$ is $s(t)=\sin(t)+1$.